Ballymena and Omagh, who both received a bye in the first round, will make their competition bow against each other at Eaton Park.
Malone, who comfortably saw off the challenge of Ulster Bank League Division 2B side City of Derry, will now play fellow 2B outfit Dungannon.
Holders Ballynahinch, who scraped past Belfast Harlequins in the opening round, have been handed another away trip to Bangor.
Banbridge's reward for a win over Rainey is a trip to the Palace Grounds to take on City of Armagh.
First Trust Senior Cup 2017/18
Belfast Harlequins 13 - 18 Ballynahinch
Malone 47 - 3 City Of Derry
Queen's University 15 - 20 Dungannon
Rainey OB 12 - 32 Banbridge
Quarter-Finals (Saturday 14th October)
Ballymena v Omagh
Updated 14:49 - 22 Sep 2017 by Stephen Barbour
Malone v Dungannon
Bangor v Ballynahinch
City of Armagh v Banbridge